Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

Martini Workshop 2015

Coarse Graining Basics

Alex de Vries Every word or concept, clear as it may seem to be,

has only a limited range of applicability

Werner Heisenberg

Recent MARTINI work› Mammalian membrane simulation of

unprecedented complexity and duration.

Ingólfsson et al. J. Amer. Chem. Soc. 136, 14554 (2014)

• 63 different lipids, asymmetric layers.

• ~0.5 Mbeads, 40 μs• enables the high-

resolution study of the distribution and dynamics of lipids interacting with each other.

• the next step is to add proteins.

Ingredients for Molecular Modeling

Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

ENERGY

MOTION

BUILDING BLOCKS

SYSTEM

Popular Molecular Models› All Atoms (OPLS-AA/L,

CHARMM, AMBER)• Each atom is treated as a particle

› United Atoms (GROMOS)• Each atom is treated as a particle,

except aliphatic CH, CH2, CH3

› Coarse grained superatoms (MARTINI)• A group of (~4 for Martini) united

atoms is treated as a particle› DPPC molecule (a lipid) in UA and

CG (MARTINI) representations

Common Potentials in Molecular Models› BONDED: Simple harmonic/cosine bonds & angles

V φ( ) = Kn 1+ cos nφ +δ( )⎡⎣ ⎤⎦n∑V d( ) = k

2d − d0( )2 V θ( ) = kθ

2θ −θ0( )2

V d( ) = k4d 2 − d0

2( )2 V θ( ) = kθ2cosθ − cosθ0( )2

*

*

*

* type used in standard Martini

› Usually simple functions with some physical justification

› NON-BONDED: Coulomb interaction between partial charges (electrostatics) and Lennard-Jones potential (long-range dispersion and short-range repulsion)

› distance ›

ener

gy

› ε determines strength of the interaction

› σ determines optimal distance of the interaction ECoulomb =

q1q2r12

ELJ = Edispersion + Erepulsion

= 4ε −σ6

r126 + σ 12

r1212

⎛⎝⎜

⎞⎠⎟

Common Potentials in Molecular Models

› Full descriptions of force fields implemented in GROMACS are described in the manual, Chapter 4

* Martini uses modified, smoothed variants of these!

| 10Basic MD Algorithm› Numerically integrate Newton’s Law: F = m*a

› t=0

› t=1

› Forces & velocities

› displacements

› new displacements, etc...

ma = mr = F r( ) = −∇E r( ) Δr = rΔt + F

2mΔt( )2

A Very Simple System

2 particles

vacuum

constant E

V R( ) = 12k R −R0( )2

› What goes in...

› Single oscillator treated classically in vacuo

› Single oscillator treated classically in vacuo

A Very Simple System

coordinates

vacuum

constant E

V R( ) = 12k R −R0( )2

› What goes in... using Gromacs

.gro or .pdb

.top or .itp

.mdp

also algorithm for integration, number of steps, time step Δt, how often should we write output, and many other options!

definition of the system and of the molecules and their bonded and nonbonded interactions

Adapted from Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

.top

.itp.gro

.mdp

grompp mdrun.tprAnalysis uses g_... and scripts on .xtc, .trr, and .edr

CHECK!!!!

GROMACS .files and programs

› Descriptions of the system• Positions

• Velocities• Forces

• Energies• Other system information

Time-traces

› What comes out... using Gromacs

.edr

.xtc and/or.trr

› The user must then extract this information and visualize it• Gromacs has many tools for this!• Always VISUALLY inspect your simulation

(ngmx, VMD, Pymol, ...)

› Time-trace data can extracted and visualized

Visualization of some results

g_bond

Bond distance and relative velocity as a function of time

› Time-trace data can be collected in a distribution

Sampling a Distribution

Distribution of distances in harmonic oscillator model

g_bond

Bond distance and relative velocity as a function of time

› Relevant or interesting degrees of freedom› Harmonic oscillator traces out a simple path

described by• Distance• Relative velocity

Useful and Informative Description(s)

› What comes out...

Distance (d) versus relative velocity (v) for a simple harmonic oscillator.

For an oscillator at constant energy, the ellipse constitutes its phase space. If the energy is not constant, many ellipses span its phase space, characterized by different amplitudes. (The path is traversed in counter clock-wise fashion.)

› Describes possible states of the system• Dimension 2*Ndof (R,P) or Ndof (R)

• A single system, in time, follows an allowed path through phase space visiting allowed states (R,P)

A Tasting (Sampling) of Classical Phase Space

dof=degree(s) of freedom

› What comes out...

› The Boltzmann distribution is not observed for a single harmonic oscillator in vacuo

Potential and Distribution

p r( )∝???e−E r( )kBT = e

−k r−r0( )22kBTE r( ) = k

2r − r0( )2

› The Boltzmann distribution is observed for anharmonically coupled oscillators

Potential and Distribution

p r( )∝!!!e−k r−r0( )22kBT

› Close link between potential and distribution

• Can be used to extract potential from structural data

• Inverse Boltzmann techniques• In MARTINI for fine-tuning bonded interactions

Potential and Distribution

p r( ) = e−E r( )kBT

Q⇔ E r( ) = −kBT ln p r( )Q⎡⎣ ⎤⎦

p r( )E r( )

r→

Q = e− EikBT∑ → Z = e

−E r( )kBT dr∫

› Define states as regions in phase space• Several realizations in

the same state• Free energy difference

between two states is related to the ratio of the number of realizations in each state and/or the probabilities of finding the system in a conformation belonging to a state

› The folded state of a protein is not a single conformation

What we want to sample phase space for

K12 =p2eq

p1eq =

e−βEii∈2∑e−βEj

j∈1∑

= e−βΔG120

1 23

β = 1kBT

Sampling: a Problem?!

Pictures from presentation by Sander Pronk and Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

› (Free) energy landscape is complicated and huge

› Obtaining a complete picture of the possible conformations and their probability of occurring is hampered by the sheer number of them and the barriers between local minima in the (free) energy landscape

Enhanced Sampling Opportunities

Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

Coarse-graining: reducing the number of degrees of freedom, preserving the relevant physics

Biasing: adapting interactions to reduce phase space and/or smoothen the free energy landscape

Jumping: exchanging snapshots between conditions to overcome barriers

Multiscaling: reducing detail in the surroundings leading to effective interactions

Enhanced Sampling Opportunities

Coarse-graining: reducing the number of degrees of freedom, preserving the relevant physics

Multiscaling: reducing detail in the surroundings leading to effective interactions

Biasing: adapting interactions to reduce phase space and/or smoothen the free energy landscape

Jumping: exchanging snapshots between conditions to overcome barriers

Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

THERE IS NO SUCH THING AS A FREE LUNCH!

COARSE GRAINING BENEFITS› Reduced Complexity

• Physics: not all detail is relevant for our question

› Efficiency: increase length and time scales• Space: reduced density reduces number of

interactions, e.g. in 4-to-1 mapping• 4 (number) x 4 (pairs)• 4 neighbor searching

• Time: smoother energy landscape, increased time step• in the algorithm: 10-20• effective time: 4

• Total: 2.5-5 103 speed-up

Martini Workshop 2015

COARSE GRAINING HowTo

Alex de Vries

A Simple(?) Example

Coarse-graining Caveats

Coarse-graining Philosophies

A SIMPLE(?) EXAMPLE› 2-to-1 mapping scheme unto simple oscillator

• Weak coupling through collisions (gas)

› Harmonic springs

› Small mass

› Large mass

› Weak LJ potential

› 2-to-1 mapping scheme unto simple oscillator• Coarse grain on two centers of mass

A SIMPLE(?) EXAMPLE

› THE REFERENCE• A look at 100 ps

trajectory for the system (looped)

• Energy exchange through collisions

• Run at constant Energy

A SIMPLE(?) EXAMPLE

› COARSE-GRAINED BEHAVIOR• A look at 10 ps trajectory for one oscillator

A SIMPLE(?) EXAMPLE

Distance between CG centers after 2-to-1 mapping

› COLLECT DISTRIBUTIONS OF INTEREST• Distribution reflects effective interaction

• Can possibly be achieved by a simple potential

A SIMPLE(?) EXAMPLE

› Harmonic force constant from normalized Gaussian distribution

› Fit for best r0 and σ gives:

DERIVING POTENTIAL

p r( )∝ e−E r( )kBT ∝ e

−k r−r0( )22kBT

1σ π

e−r−r0( )2σ 2

dr∫ = 1

k = 2kBTσ 2

E r( ) = k2r − r0( )2

p r( ) = e−E r( )kBT

Q⇔ E r( ) = −kBT ln p r( )Q⎡⎣ ⎤⎦

› Harmonic force constant from Gaussian distribution

› Fit for best r0 and σ:

DERIVING POTENTIAL

E r( ) = k2r − r0( )2

p r( )∝ e−k r−r0( )22kBT

k = 2kBTσ 2

k = 1,269kJ ⋅mol-1 ⋅nm-2

r0 = 0.1185nm

COARSE GRAINING CAVEATS› Compare distributions at CG level

• In this simple(?) example, there are already some complications

• Frequent collisions required to get good statistics

• Need to account for rotation that increases the bond length because of centrifugal force (correlations between degrees of freedom)

• Must use r0 ≈ 0.11 nm in CG model instead of the 0.1185 nm found by fitting

• K and r0 will depend on total energy (temperature)

W.G. Noid et al, e.g. C.R. Ellis et al. Macromol. Theory Simul. 20, 478 (2011)

COARSE GRAINING CAVEATS› Smoother interaction, smoother motion

› Original, mapped › Coarse-grained

COARSE GRAINING CAVEATS

› THE MEANING OF TIME• Smoother interaction, smoother motion

• Enables larger time steps• Friction is lower, sampling speeds up• Barriers are more easily overcome

› DETAIL IS LOST• Physics may be different• Need to be careful in interpreting dynamics • Exchange of energy between modes may be less

efficient

COARSE GRAINING CAVEATS› THE MEANING OF TIME

• Re-introduce friction through stochastic term • Part of benefit is gone, but dynamics may appear

more realistic

Δr = rΔt + F

2mΔt( )2 + ξ

Coarse-graining Philosophies› HIERARCHICAL MODELING

• From quantum mechanics to evolution of galaxies• Interactions at less detailed level are the result of

the collective interactions at more detailed level• General method applicable to any system (like an

algorithm)

› EFFICIENT MODEL AT CERTAIN SCALE • Reproduce faithfully certain chosen properties

• Developed with certain application in mind• Nevertheless aiming at wide use through

considering the physics of the problems in mind

H.J.C. Berendsen Simulating the Physical World Cambridge University Press (2007)

Hierarchical modeling: pros and cons› PRO

• UNBIASED• Physics follows through the hierarchy of models

• STRAIGHTFORWARD MULTISCALING• Enables reliable combination of levels of modeling• Entirely general approach

› CON• REQUIRES LARGE WORK LOAD

• Need detailed level simulations to derive CG potentials• Complicated numerical potentials

• LIMITED VALIDITY• Strictly valid for one state point only (new system, new

potentials)

Semi-empirical modeling: pros and cons› PRO

• CHEAP• Parameterize on empirical data available• Simple analytical potentials

• TRANSFERABLE• After parameterizing building blocks, many similar systems

can be treated straightforwardly

› CON• BIASED

• Toward parameterized properties

• PROBLEMATIC MULTISCALING• Different levels do not need to correspond closely• Extensive validation required

MARTINI MODEL IS A SEMI-EMPIRICAL FORCE FIELD

› For an empirical CG model • Results of complex systems should be consistent

with experiment even though not explicitly parameterized on that complex system

› MARTINI does pretty well for lipids• Many examples are described in literature

› MARTINI extensions• Proteins• Sugars• Polymers• ...

What do we expect of the Martini CG model?

Marrink and Tieleman Chem. Soc. Rev. 42, 6801 (2013)Check out http://www.cgmartini.nl

Example: liquid hexadecane

› Standard Martini model* compared to mapped GROMOS 53A6 model

› Movie shows CG representation and is looping over 0.2 ns

› 320 hexadecane molecules in the system

› T = 300 K

› * Parameterization discussed in the next lecture

Example: liquid hexadecane

› Movie is looping over 5 ns

› 1 hexadecane molecule 0f 320 in the system

› Overall translation and rotation of the molecule is removed

Characterizing a collection of structures

› Are conformations realistic?› Time scale of sampling

› Schlitter’s formula for configurational entropy• Upper bound• Approximation for harmonic oscillator

› Procedure• Fit (part of) the structure to remove

translation (and rotation)

J. Schlitter Chem. Phys. Lett. 215, 617 (1993)

Strue ≤ S =

kB2ln 1+ kBTe

2

2D

Mass-weighted covariance matrix

Mass-weighted covariance matrix Strue ≤ S =

kB2ln 1+ kBTe

2

2D

D =

m1

Kr1k − r1( )2

k∑ …

1K

r1k − r1( ) rNk − rN( )

k∑

1K

rNk − rN( ) r1k − r1( )

k∑

mN

KrNk − rN( )2

k∑

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

r1k

m1

› Here, K is the number of conformations› is the position of atom/bead number 1 in frame k › is the mass of atom/bead 1› Note similarity to variance!

Baron et al J. Phys. Chem. B 110, 15602 (2006)

› MARTINI versus GROMOS 53A6• Studied at FG (AL) and CG levels by Baron et al.

Configurational Entropy

Baron et al J. Phys. Chem. B 110, 8464 (2006)

› Mapped FG

Strue ≤ S =

kB2ln 1+ kBTe

2

2D

|

› For phospholipid DPPC in bilayer

Configurational Entropy: Time Scale Strue ≤ S =

kB2ln 1+ kBTe

2

2D

Average structures

Baron et al. J. Phys. Chem. B 110, 15602 (2006)

Build-up of sampling after mapping

› MARTINI versus GROMOS 53A6• Bonded distributions MARTINI and mapped FG differ

Bonded interactions: compare to FG model

Baron et al J. Phys. Chem. B 110, 8464 (2006)

› Mapped FG

• Some conformations of hexadecane are more favorable in FG because of short contacts but overlapping vdW spheres in CG

COARSE GRAINING SUMMARY› SPEEDING UP SEARCH IN PHASE SPACE

• Use an effective interaction from distribution

• Smooths Potential Energy Surface• Reduces number of interactions• Increases effective time

› DETAIL IS LOST• Beneficial: detail may not be required• Detrimental: some system characteristics are gone

Thank you for your attention

› To compute a new position after a time interval ∆t

› Similarly, for the new momentum

› Combining these from t = 0

Integrating equations of motion

pi t( ) = midri t( )dt

; Fi t( ) = dpi t( )dt

ri t + Δt( ) = ri t( ) + pi τ( )mi

dτt

t+Δt

∫ ≈ ri t( ) + pi t( )mi

Δt

pi t + Δt( ) = pi t( ) + Fi τ( )dτt

t+Δt

∫ ≈ pi t( ) + Fi t( )Δt

ri Δt( ) = ri 0( ) + 1mi

pi 0( ) + Fi ′τ( )d ′τ0

τ

∫⎡⎣⎢

⎤⎦⎥dτ

0

Δt

∫

≈ ri 0( ) + pi 0( )mi

Δt + Fi 0( )2mi

Δt( )2

Verlet Integrator

Taylor expansion of the position in time, assuming constant force

F = ma = m dvdt

= m ddt

drdt

⎛⎝⎜

⎞⎠⎟ = m

d 2rdt 2

r t + Δt( ) = r t( ) + drdt t

Δt + 12!d 2rdt 2 t

Δt( )2 + ...

r t − Δt( ) = r t( )− drdt t

Δt + 12!d 2rdt 2 t

Δt( )2 − ...

r t + Δt( ) + r t − Δt( ) = 2r t( ) + d2rdt 2 t

Δt( )2 +O( Δt( )4 )

⇒ r t + Δt( ) + r t − Δt( ) ≈ 2r t( ) + Fm

Δt( )2

F = m d 2rdt 2

r t + Δt( )− r t( ) ≈r t( )− r t − Δt( )

+ Fm

Δt( )2r t + Δt( )− r t( )

Fm

Δt( )2

r t( )− r t − Δt( )r t( )

r t − Δt( )r t + Δt( )

Verlet integrator from Newton’s Law

r t + Δt( )− r t( )⎡⎣ ⎤⎦ − r t( )− r t − Δt( )⎡⎣ ⎤⎦ =F t( )m

Δt 2

is the difference of the difference, or the change in the changed 2rdt 2

Conservation of Energy› In practice, even with small time-steps,

conservation of energy is difficult to achieve

ΔE = −W = −F•ΔR

.mdp

ΔE = −W = − F• dR∫

Too large a time-step takes the system away from its potential energy surface and leads to failure of energy conservation.

Practical considerations: Conservation of Energy› In numerical approaches, time-step should be such that

PES is followed

› We should end up at the potential energy we expect or energy is lost from/added to the system

ΔEpred ≈ −F r( )• F r( ) Δt( )22m

Δr ≈F r( ) Δt( )22m

ΔE = −F r( )•Δr ma = mr = F r( ) = −∇E r( )

Δr = rΔt + F

2mΔt( )2

r = v = dr

dt; r = a = d

2rdt 2E r( ) U r( )and are used interchangeably

|

› At atomistic level, time step for fastest vibrations

• Time step ~1 fs

Appropriate time step for sampling

EtotR

M

p R( )

1 / Δtvib = ν IR = cvIR ≈

3*108 * 3,000 ms−1cm−1⎡⎣ ⎤⎦≈ 1*1014Hz

Cut-off noise› Particles start/stop interacting at certain distance:

energy is not conserved

RcRc

› In practice, energy conservation is impossible to achieve

ma = mr = F r( ) = −∇E r( )

Δr = rΔt + F

2mΔt( )2

Practical considerations: Conservation of Energy

› Couple to heat bath to dissipate or gain heat› Popular methods are velocity rescaling (e.g.

Berendsen) or extended ensembles (e.g. Nose-Hoover)› Strictly speaking, we are not doing Newtonian

mechanics› Time step and bath coupling strengths are part of the

parameter set!› Martini uses non-bonded potentials without jump at

cut-off (shifted potentials)